\(S_1\)  :
The SU(3) flavor symmetry parameters
without the baryonic momentum-transfer \(q^2\)  dependence. We treat the SU(3) flavor parameters \((E)^{L(R),s_p,s_k}_{VA,\lambda }(q^2)\)  as constants without \(q^2\)  dependence, which will lead \(H_M(q^2)\)  in Eq. (REF ) to a constant, too.
We use the \(1\sigma \)  error experimental data of \(\mathcal {B}(\Lambda ^0_b\rightarrow \Lambda ^0 \mu ^+\mu ^-)\)  to constrain \(H_M(q^2)\)  (\(i.e.\) , \(|E|^2\) ), and then predict other \(\mathcal {B}(T_{b3}\rightarrow T_8\ell ^+\ell ^-)\)  by the amplitude relations in Tab. REF .
\(S_2\) :
The SU(3) flavor symmetry parameters with the baryonic momentum-transfer \(q^2\)  dependence. In order to obtain more precise predictions, we use
the hadronic helicity amplitude expressions in Eq. (REF ), which are \(q^2\)  dependent and can be expressed by the Wilson coefficients and the form factors.
The expressions of the Wilson coefficients without the LD contributions are taken from Ref. [1]}.
As for the \(q^2\)  dependent form factors involving the
\(T_{b3}\rightarrow T_8\)  transitions, we use the recent lattice QCD results of \(\Lambda ^0_b\rightarrow \Lambda ^0\)  [2]}, in which the form factors are parametrized by
\(f(q^2)=\frac{f(0)}{1-q^2/(m^f_{pole})^2}\left[1+\frac{a^f_1}{a^f_0}z(q^2)+\frac{a^f_2}{a^f_0}[z(q^2)]^2\right],\) 
where \(f=f_+,f_\perp ,f_0,g_+,g_\perp ,g_0,h_+,h_\perp ,\widetilde{h}_+,\widetilde{h}_\perp \) , and the details of \(z(q^2)\)  and \(m^f_{pole}\)  can be found in Ref. [2]}. We keep \(f_+(0)\)  as an undetermined constant without \(q^2\)  dependence, and other \(f(0)\)  can be expressed as \(\frac{a^{f}_0 }{a^{f_+}_0}f_+(0)\) . The central values of \(a^f_i\)  in Tab. V of Ref. [2]} will be used in our analysis.
Since these form factors also preserve the SU(3) flavor symmetry, the same relations in Tab. REF  will be used for \(f_+(0)\) . We use the \(1\sigma \)  error experimental data of \(\mathcal {B}(\Lambda ^0_b\rightarrow \Lambda ^0 \mu ^+\mu ^-)\)  to constrain \(f_+(0)\)  and then predict other \(\mathcal {B}(T_{b3}\rightarrow T_8\ell ^+\ell ^-)\) .

